\(\int F^{c (a+b x)} \cosh ^2(d+e x) \, dx\) [286]

Optimal result

Integrand size = 18, antiderivative size = 132 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {b c F^{c (a+b x)} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)} \]

[Out]

2*e^2*F^(c*(b*x+a))/b/c/ln(F)/(4*e^2-b^2*c^2*ln(F)^2)-b*c*F^(c*(b*x+a))*cosh(e*x+d)^2*ln(F)/(4*e^2-b^2*c^2*ln(
F)^2)+2*e*F^(c*(b*x+a))*cosh(e*x+d)*sinh(e*x+d)/(4*e^2-b^2*c^2*ln(F)^2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5585, 2225} \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=-\frac {b c \log (F) \cosh ^2(d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e \sinh (d+e x) \cosh (d+e x) F^{c (a+b x)}}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )} \]

[In]

Int[F^(c*(a + b*x))*Cosh[d + e*x]^2,x]

[Out]

(2*e^2*F^(c*(a + b*x)))/(b*c*Log[F]*(4*e^2 - b^2*c^2*Log[F]^2)) - (b*c*F^(c*(a + b*x))*Cosh[d + e*x]^2*Log[F])
/(4*e^2 - b^2*c^2*Log[F]^2) + (2*e*F^(c*(a + b*x))*Cosh[d + e*x]*Sinh[d + e*x])/(4*e^2 - b^2*c^2*Log[F]^2)

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 5585

Int[Cosh[(d_.) + (e_.)*(x_)]^(n_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a +
b*x))*(Cosh[d + e*x]^n/(e^2*n^2 - b^2*c^2*Log[F]^2)), x] + (Dist[n*(n - 1)*(e^2/(e^2*n^2 - b^2*c^2*Log[F]^2))
, Int[F^(c*(a + b*x))*Cosh[d + e*x]^(n - 2), x], x] + Simp[e*n*F^(c*(a + b*x))*Sinh[d + e*x]*(Cosh[d + e*x]^(n
- 1)/(e^2*n^2 - b^2*c^2*Log[F]^2)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*n^2 - b^2*c^2*Log[F]^2, 0]
&& GtQ[n, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {b c F^{c (a+b x)} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {\left (2 e^2\right ) \int F^{c (a+b x)} \, dx}{4 e^2-b^2 c^2 \log ^2(F)} \\ & = \frac {2 e^2 F^{c (a+b x)}}{b c \log (F) \left (4 e^2-b^2 c^2 \log ^2(F)\right )}-\frac {b c F^{c (a+b x)} \cosh ^2(d+e x) \log (F)}{4 e^2-b^2 c^2 \log ^2(F)}+\frac {2 e F^{c (a+b x)} \cosh (d+e x) \sinh (d+e x)}{4 e^2-b^2 c^2 \log ^2(F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.64 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\frac {F^{c (a+b x)} \left (-4 e^2+b^2 c^2 \log ^2(F)+b^2 c^2 \cosh (2 (d+e x)) \log ^2(F)-2 b c e \log (F) \sinh (2 (d+e x))\right )}{-8 b c e^2 \log (F)+2 b^3 c^3 \log ^3(F)} \]

[In]

Integrate[F^(c*(a + b*x))*Cosh[d + e*x]^2,x]

[Out]

(F^(c*(a + b*x))*(-4*e^2 + b^2*c^2*Log[F]^2 + b^2*c^2*Cosh[2*(d + e*x)]*Log[F]^2 - 2*b*c*e*Log[F]*Sinh[2*(d +
e*x)]))/(-8*b*c*e^2*Log[F] + 2*b^3*c^3*Log[F]^3)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.68

method	result	size
parallelrisch	\(-\frac {2 F^{c \left (b x +a \right )} \left (-\frac {c^{2} b^{2} \ln \left (F \right )^{2} \cosh \left (2 e x +2 d \right )}{2}-\frac {b^{2} c^{2} \ln \left (F \right )^{2}}{2}+\ln \left (F \right ) b c e \sinh \left (2 e x +2 d \right )+2 e^{2}\right )}{2 \ln \left (F \right )^{3} b^{3} c^{3}-8 \ln \left (F \right ) b c \,e^{2}}\)	\(90\)
risch	\(\frac {\left (\ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{4 e x +4 d}+2 \ln \left (F \right )^{2} b^{2} c^{2} {\mathrm e}^{2 e x +2 d}-2 \ln \left (F \right ) b c e \,{\mathrm e}^{4 e x +4 d}+b^{2} c^{2} \ln \left (F \right )^{2}+2 b c \ln \left (F \right ) e -8 e^{2} {\mathrm e}^{2 e x +2 d}\right ) {\mathrm e}^{-2 e x -2 d} F^{c \left (b x +a \right )}}{4 b c \ln \left (F \right ) \left (b c \ln \left (F \right )-2 e \right ) \left (b c \ln \left (F \right )+2 e \right )}\)	\(143\)

[In]

int(F^(c*(b*x+a))*cosh(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

-2*F^(c*(b*x+a))*(-1/2*c^2*b^2*ln(F)^2*cosh(2*e*x+2*d)-1/2*b^2*c^2*ln(F)^2+ln(F)*b*c*e*sinh(2*e*x+2*d)+2*e^2)/
(2*ln(F)^3*b^3*c^3-8*ln(F)*b*c*e^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 699 vs. \(2 (128) = 256\).

Time = 0.27 (sec) , antiderivative size = 699, normalized size of antiderivative = 5.30 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\frac {{\left ({\left (b^{2} c^{2} \log \left (F\right )^{2} - 2 \, b c e \log \left (F\right )\right )} \sinh \left (e x + d\right )^{4} - 8 \, e^{2} \cosh \left (e x + d\right )^{2} + 4 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \left (F\right )^{2} - 2 \, b c e \cosh \left (e x + d\right ) \log \left (F\right )\right )} \sinh \left (e x + d\right )^{3} + {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{4} + 2 \, b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (6 \, b c e \cosh \left (e x + d\right )^{2} \log \left (F\right ) - {\left (3 \, b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \left (F\right )^{2} + 4 \, e^{2}\right )} \sinh \left (e x + d\right )^{2} - 2 \, {\left (b c e \cosh \left (e x + d\right )^{4} - b c e\right )} \log \left (F\right ) - 4 \, {\left (2 \, b c e \cosh \left (e x + d\right )^{3} \log \left (F\right ) + 4 \, e^{2} \cosh \left (e x + d\right ) - {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{3} + b^{2} c^{2} \cosh \left (e x + d\right )\right )} \log \left (F\right )^{2}\right )} \sinh \left (e x + d\right )\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + {\left ({\left (b^{2} c^{2} \log \left (F\right )^{2} - 2 \, b c e \log \left (F\right )\right )} \sinh \left (e x + d\right )^{4} - 8 \, e^{2} \cosh \left (e x + d\right )^{2} + 4 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \left (F\right )^{2} - 2 \, b c e \cosh \left (e x + d\right ) \log \left (F\right )\right )} \sinh \left (e x + d\right )^{3} + {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{4} + 2 \, b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (6 \, b c e \cosh \left (e x + d\right )^{2} \log \left (F\right ) - {\left (3 \, b^{2} c^{2} \cosh \left (e x + d\right )^{2} + b^{2} c^{2}\right )} \log \left (F\right )^{2} + 4 \, e^{2}\right )} \sinh \left (e x + d\right )^{2} - 2 \, {\left (b c e \cosh \left (e x + d\right )^{4} - b c e\right )} \log \left (F\right ) - 4 \, {\left (2 \, b c e \cosh \left (e x + d\right )^{3} \log \left (F\right ) + 4 \, e^{2} \cosh \left (e x + d\right ) - {\left (b^{2} c^{2} \cosh \left (e x + d\right )^{3} + b^{2} c^{2} \cosh \left (e x + d\right )\right )} \log \left (F\right )^{2}\right )} \sinh \left (e x + d\right )\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{4 \, {\left (b^{3} c^{3} \cosh \left (e x + d\right )^{2} \log \left (F\right )^{3} - 4 \, b c e^{2} \cosh \left (e x + d\right )^{2} \log \left (F\right ) + {\left (b^{3} c^{3} \log \left (F\right )^{3} - 4 \, b c e^{2} \log \left (F\right )\right )} \sinh \left (e x + d\right )^{2} + 2 \, {\left (b^{3} c^{3} \cosh \left (e x + d\right ) \log \left (F\right )^{3} - 4 \, b c e^{2} \cosh \left (e x + d\right ) \log \left (F\right )\right )} \sinh \left (e x + d\right )\right )}} \]

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,x, algorithm="fricas")

[Out]

1/4*(((b^2*c^2*log(F)^2 - 2*b*c*e*log(F))*sinh(e*x + d)^4 - 8*e^2*cosh(e*x + d)^2 + 4*(b^2*c^2*cosh(e*x + d)*l
og(F)^2 - 2*b*c*e*cosh(e*x + d)*log(F))*sinh(e*x + d)^3 + (b^2*c^2*cosh(e*x + d)^4 + 2*b^2*c^2*cosh(e*x + d)^2
+ b^2*c^2)*log(F)^2 - 2*(6*b*c*e*cosh(e*x + d)^2*log(F) - (3*b^2*c^2*cosh(e*x + d)^2 + b^2*c^2)*log(F)^2 + 4*
e^2)*sinh(e*x + d)^2 - 2*(b*c*e*cosh(e*x + d)^4 - b*c*e)*log(F) - 4*(2*b*c*e*cosh(e*x + d)^3*log(F) + 4*e^2*co
sh(e*x + d) - (b^2*c^2*cosh(e*x + d)^3 + b^2*c^2*cosh(e*x + d))*log(F)^2)*sinh(e*x + d))*cosh((b*c*x + a*c)*lo
g(F)) + ((b^2*c^2*log(F)^2 - 2*b*c*e*log(F))*sinh(e*x + d)^4 - 8*e^2*cosh(e*x + d)^2 + 4*(b^2*c^2*cosh(e*x + d
)*log(F)^2 - 2*b*c*e*cosh(e*x + d)*log(F))*sinh(e*x + d)^3 + (b^2*c^2*cosh(e*x + d)^4 + 2*b^2*c^2*cosh(e*x + d
)^2 + b^2*c^2)*log(F)^2 - 2*(6*b*c*e*cosh(e*x + d)^2*log(F) - (3*b^2*c^2*cosh(e*x + d)^2 + b^2*c^2)*log(F)^2 +
4*e^2)*sinh(e*x + d)^2 - 2*(b*c*e*cosh(e*x + d)^4 - b*c*e)*log(F) - 4*(2*b*c*e*cosh(e*x + d)^3*log(F) + 4*e^2
*cosh(e*x + d) - (b^2*c^2*cosh(e*x + d)^3 + b^2*c^2*cosh(e*x + d))*log(F)^2)*sinh(e*x + d))*sinh((b*c*x + a*c)
*log(F)))/(b^3*c^3*cosh(e*x + d)^2*log(F)^3 - 4*b*c*e^2*cosh(e*x + d)^2*log(F) + (b^3*c^3*log(F)^3 - 4*b*c*e^2
*log(F))*sinh(e*x + d)^2 + 2*(b^3*c^3*cosh(e*x + d)*log(F)^3 - 4*b*c*e^2*cosh(e*x + d)*log(F))*sinh(e*x + d))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (119) = 238\).

Time = 1.13 (sec) , antiderivative size = 706, normalized size of antiderivative = 5.35 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\begin {cases} x \cosh ^{2}{\left (d \right )} & \text {for}\: F = 1 \wedge b = 0 \wedge c = 0 \wedge e = 0 \\- \frac {x \sinh ^{2}{\left (d + e x \right )}}{2} + \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e} & \text {for}\: F = 1 \\F^{a c} \left (- \frac {x \sinh ^{2}{\left (d + e x \right )}}{2} + \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e}\right ) & \text {for}\: b = 0 \\- \frac {x \sinh ^{2}{\left (d + e x \right )}}{2} + \frac {x \cosh ^{2}{\left (d + e x \right )}}{2} + \frac {\sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{2 e} & \text {for}\: c = 0 \\\frac {F^{a c + b c x} x \sinh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{4} - \frac {F^{a c + b c x} x \sinh {\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )} \cosh {\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{2} + \frac {F^{a c + b c x} x \cosh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{4} - \frac {F^{a c + b c x} \sinh {\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )} \cosh {\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{2 b c \log {\left (F \right )}} + \frac {F^{a c + b c x} \cosh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} - d \right )}}{b c \log {\left (F \right )}} & \text {for}\: e = - \frac {b c \log {\left (F \right )}}{2} \\\frac {F^{a c + b c x} x \sinh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{4} - \frac {F^{a c + b c x} x \sinh {\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )} \cosh {\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{2} + \frac {F^{a c + b c x} x \cosh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{4} - \frac {F^{a c + b c x} \sinh {\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )} \cosh {\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{2 b c \log {\left (F \right )}} + \frac {F^{a c + b c x} \cosh ^{2}{\left (\frac {b c x \log {\left (F \right )}}{2} + d \right )}}{b c \log {\left (F \right )}} & \text {for}\: e = \frac {b c \log {\left (F \right )}}{2} \\\frac {F^{a c + b c x} b^{2} c^{2} \log {\left (F \right )}^{2} \cosh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - 4 b c e^{2} \log {\left (F \right )}} - \frac {2 F^{a c + b c x} b c e \log {\left (F \right )} \sinh {\left (d + e x \right )} \cosh {\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - 4 b c e^{2} \log {\left (F \right )}} + \frac {2 F^{a c + b c x} e^{2} \sinh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - 4 b c e^{2} \log {\left (F \right )}} - \frac {2 F^{a c + b c x} e^{2} \cosh ^{2}{\left (d + e x \right )}}{b^{3} c^{3} \log {\left (F \right )}^{3} - 4 b c e^{2} \log {\left (F \right )}} & \text {otherwise} \end {cases} \]

[In]

integrate(F**(c*(b*x+a))*cosh(e*x+d)**2,x)

[Out]

Piecewise((x*cosh(d)**2, Eq(F, 1) & Eq(b, 0) & Eq(c, 0) & Eq(e, 0)), (-x*sinh(d + e*x)**2/2 + x*cosh(d + e*x)*
*2/2 + sinh(d + e*x)*cosh(d + e*x)/(2*e), Eq(F, 1)), (F**(a*c)*(-x*sinh(d + e*x)**2/2 + x*cosh(d + e*x)**2/2 +
sinh(d + e*x)*cosh(d + e*x)/(2*e)), Eq(b, 0)), (-x*sinh(d + e*x)**2/2 + x*cosh(d + e*x)**2/2 + sinh(d + e*x)*
cosh(d + e*x)/(2*e), Eq(c, 0)), (F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/2 - d)**2/4 - F**(a*c + b*c*x)*x*sinh(b*
c*x*log(F)/2 - d)*cosh(b*c*x*log(F)/2 - d)/2 + F**(a*c + b*c*x)*x*cosh(b*c*x*log(F)/2 - d)**2/4 - F**(a*c + b*
c*x)*sinh(b*c*x*log(F)/2 - d)*cosh(b*c*x*log(F)/2 - d)/(2*b*c*log(F)) + F**(a*c + b*c*x)*cosh(b*c*x*log(F)/2 -
d)**2/(b*c*log(F)), Eq(e, -b*c*log(F)/2)), (F**(a*c + b*c*x)*x*sinh(b*c*x*log(F)/2 + d)**2/4 - F**(a*c + b*c*
x)*x*sinh(b*c*x*log(F)/2 + d)*cosh(b*c*x*log(F)/2 + d)/2 + F**(a*c + b*c*x)*x*cosh(b*c*x*log(F)/2 + d)**2/4 -
F**(a*c + b*c*x)*sinh(b*c*x*log(F)/2 + d)*cosh(b*c*x*log(F)/2 + d)/(2*b*c*log(F)) + F**(a*c + b*c*x)*cosh(b*c*
x*log(F)/2 + d)**2/(b*c*log(F)), Eq(e, b*c*log(F)/2)), (F**(a*c + b*c*x)*b**2*c**2*log(F)**2*cosh(d + e*x)**2/
(b**3*c**3*log(F)**3 - 4*b*c*e**2*log(F)) - 2*F**(a*c + b*c*x)*b*c*e*log(F)*sinh(d + e*x)*cosh(d + e*x)/(b**3*
c**3*log(F)**3 - 4*b*c*e**2*log(F)) + 2*F**(a*c + b*c*x)*e**2*sinh(d + e*x)**2/(b**3*c**3*log(F)**3 - 4*b*c*e*
*2*log(F)) - 2*F**(a*c + b*c*x)*e**2*cosh(d + e*x)**2/(b**3*c**3*log(F)**3 - 4*b*c*e**2*log(F)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + 2 \, e x + 2 \, d\right )}}{4 \, {\left (b c \log \left (F\right ) + 2 \, e\right )}} + \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - 2 \, e x\right )}}{4 \, {\left (b c e^{\left (2 \, d\right )} \log \left (F\right ) - 2 \, e e^{\left (2 \, d\right )}\right )}} + \frac {F^{b c x + a c}}{2 \, b c \log \left (F\right )} \]

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,x, algorithm="maxima")

[Out]

1/4*F^(a*c)*e^(b*c*x*log(F) + 2*e*x + 2*d)/(b*c*log(F) + 2*e) + 1/4*F^(a*c)*e^(b*c*x*log(F) - 2*e*x)/(b*c*e^(2
*d)*log(F) - 2*e*e^(2*d)) + 1/2*F^(b*c*x + a*c)/(b*c*log(F))

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 889, normalized size of antiderivative = 6.73 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=\text {Too large to display} \]

[In]

integrate(F^(c*(b*x+a))*cosh(e*x+d)^2,x, algorithm="giac")

[Out]

(2*b*c*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)*log(abs(F))/(4*b^2*c^2*log(ab
s(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2
*pi*a*c*sgn(F) + 1/2*pi*a*c)/(4*b^2*c^2*log(abs(F))^2 + (pi*b*c*sgn(F) - pi*b*c)^2))*e^(b*c*x*log(abs(F)) + a*
c*log(abs(F))) + I*(I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(2*I*pi*
b*c*sgn(F) - 2*I*pi*b*c + 4*b*c*log(abs(F))) - I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn
(F) + 1/2*I*pi*a*c)/(-2*I*pi*b*c*sgn(F) + 2*I*pi*b*c + 4*b*c*log(abs(F))))*e^(b*c*x*log(abs(F)) + a*c*log(abs(
F))) + 1/2*(2*(b*c*log(abs(F)) + 2*e)*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c
)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(
F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) + 2*e)^2)
)*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2*d) + I*(I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2
*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(4*I*pi*b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) + 16*e) - I*e^(-1/2*I*pi*
b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*l
og(abs(F)) + 16*e))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) + 2*e)*x + 2*d) + 1/2*(2*(b*c*log(abs(F)) - 2*e)*cos
(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*lo
g(abs(F)) - 2*e)^2) - (pi*b*c*sgn(F) - pi*b*c)*sin(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1
/2*pi*a*c)/((pi*b*c*sgn(F) - pi*b*c)^2 + 4*(b*c*log(abs(F)) - 2*e)^2))*e^(a*c*log(abs(F)) + (b*c*log(abs(F)) -
2*e)*x - 2*d) + I*(I*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(4*I*pi*
b*c*sgn(F) - 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*e) - I*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x - 1/2*I*pi*
a*c*sgn(F) + 1/2*I*pi*a*c)/(-4*I*pi*b*c*sgn(F) + 4*I*pi*b*c + 8*b*c*log(abs(F)) - 16*e))*e^(a*c*log(abs(F)) +
(b*c*log(abs(F)) - 2*e)*x - 2*d)

Mupad [B] (verification not implemented)

Time = 2.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int F^{c (a+b x)} \cosh ^2(d+e x) \, dx=-\frac {2\,F^{a\,c+b\,c\,x}\,e^2-F^{a\,c+b\,c\,x}\,b^2\,c^2\,{\mathrm {cosh}\left (d+e\,x\right )}^2\,{\ln \left (F\right )}^2+2\,F^{a\,c+b\,c\,x}\,b\,c\,e\,\mathrm {cosh}\left (d+e\,x\right )\,\mathrm {sinh}\left (d+e\,x\right )\,\ln \left (F\right )}{b^3\,c^3\,{\ln \left (F\right )}^3-4\,b\,c\,e^2\,\ln \left (F\right )} \]

[In]

int(F^(c*(a + b*x))*cosh(d + e*x)^2,x)

[Out]

-(2*F^(a*c + b*c*x)*e^2 - F^(a*c + b*c*x)*b^2*c^2*cosh(d + e*x)^2*log(F)^2 + 2*F^(a*c + b*c*x)*b*c*e*cosh(d +
e*x)*sinh(d + e*x)*log(F))/(b^3*c^3*log(F)^3 - 4*b*c*e^2*log(F))